All Questions
Tagged with plane-geometry convex-polytopes
7 questions
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Lattice points in the boundary of a Minkowski sum of two convex lattice polygons
Let $P$ and $Q$ be two convex lattice polygons in $\mathbb{R}_+^2$ and let $P+Q$ be their Minkowski sum. Given a set $S \subset \mathbb{R}^2$, we let $L(S) =\#( S \bigcap \mathbb{Z}^2)$.
The equality $...
- 183
1
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65
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Minimum bounding rectangle of symmetric convex bodies in the plane : is the ball the worst case
The minimal rectangle containing the euclidean ball in the plane is the standard cube $B_\infty = [-1;1]^2$. I would like to know if the euclidean ball is the worst symmetric convex body to be ...
- 245
7
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1
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Minimum area of the convex hull of the union of a parallelogram and a triangle
This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of ...
- 7,276
5
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Diameter-area ratio for affine tranformations.
Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$?
I found only one reference, to "Über einige ...
- 5,053
4
votes
1
answer
256
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Polar interpretation of convexity
Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...
- 645
45
votes
4
answers
5k
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Polynomial roots and convexity
A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but no one had ever ...
6
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Elementary problem about triangles inside a convex polygon
Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...
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